1. Objectives

  • Use Coulomb’s Law to determine the force between two electric charges.

  • Determine the electric field surrounding a point charge.

  • Determine the force on a charge due to an electric field.

  • Use superposition to determine the electric field due to multiple charges.

2. Notes

https://www.youtube.com/watch?v=9CXZW-7I65I — beginning to 5:50

newton v coulomb
Figure 1. Newton v. Coulomb

2.1. Coulomb’s Law

Universal Rule #1
Electric charges always exert a force on each other.

Briefly switch to Laws and Equations - Coulomb’s Law for notes on the law itself.

Click here after returning from the linked page.

Did you honestly read that section? Only you and my web server know that answer ;)

\(\epsilon_0 = 8.854 \ldots \times 10^{-12} \, \mathrm{\frac{F}{m}}\)

permittivity of free space [1]

\[\hat{a}_R = \frac{\vec{r_2} - \vec{r_1}}{\left| \vec{r_2} - \vec{r_1} \right|}\]

  • \(\vec{r_1}\) is the location of the actual/reference charge, the other one that is stationary.

  • \(\vec{r_2}\) is the location of the test charge, the one “feeling” the force.

2.2. Electric field

\[\begin{align} \mathbf{\vec{E}} &\equiv \frac{\mathbf{\vec{F}}}{q} \label{eq-e-definition}\\ &= \frac{Q}{4\pi\epsilon_0 R^2} \, \hat{a}_R \end{align}\]


Electric field vectors have units of or .


Go back to equation \(\eqref{eq-e-definition}\) and deeply contemplate this connection to the meaning of an electric field.

What is an electric field?

It is defined by the above description. Read it again slowly!

Does an E-field exist as an entity itself?

  • Can you have an E-field without a charge?

  • What about in space?

  • How can you detect the “presence” of an E-field?

Details

Much of the time it’s not useful in practice to answer these questions carefully. But, when things get weird or novel or interesting, then it pays to go back to what an (electric) vector field represents.

Chapter 4 discusses various shortcuts to finding the electric field given charge densities in various arrangements (line, loop, rectangular planar surface, planar disk, infinite plane).

  • Electric field lines start on positive charges and end on negative charges.[2]

From Example 3.2 and the definition itself, an electric field:

  • Exists at all points in 3D space.

  • Is a vector quantity at each point.

This combination is simply the definition of a vector field.

For our purposes, there can be charges spread around a volume but an elementary charge has zero volume (just a point). The physicists may need to think about the volume of a charge, but it would be numerically irrelevant to us in any case.

Why do the examples use integer coulombs when a coulomb of charge is huge?

Welcome to academic example-making! The mechanics of the calculation are the point, not necessarily getting a sense of the magnitude of typical values. Plus, it makes the calculation convenient by hand, that’s all.

3. Class time

3.1. ICE03

Complete this familiar worksheet to practice using the new formulas introduced in the lesson.

3.2. The Nickel and the Battleship

300px USS Indiana %28BB 58%29 80 G 222923
Figure 2. USS Indiana

(adapted from Power of Attraction - Dr. Howard Johnson)

Imagine a US nickel on a string hanging from a quadcopter drone. It is hovering exactly 1 km above the battleship USS Indiana cruising in the Pacfic ocean.

Remove one electron from the outer valence shell per atom for the whole nickel and transfer them to the battleship.[3]

  • How much charge is this in coulombs?

  • Convert this charge to units of  mA·h, a common unit used to describe battery capacity.

  • What is the force exerted on the USS Indiana? This is a vector, so give direction and magnitude.

  • Compare this to the (downward) force due to gravity (the ship’s weight, a.k.a. displacement[4]). Convert forces in newtons to weight on Earth in metric tons (1000 kg).

  • What is the voltage between the nickel and battleship? Voltage isn’t covered until lesson 7, so compute the Electric field, inspect the units carefully, and somehow sneak in 1 km to arrive at units of volts.

Physics and chemistry facts you need:[5]

  • Volume or mass of a nickel.

  • Atomic weight of copper and nickel, units of g/mol or grams per mole.

  • Avogadro’s number, 6.022×1023 atoms per mole.

  • Electron charge, -1.602×10-19 C.

  • Acceleration due to gravity on Earth, 9.807 m/s2.

4. Errata

Example 3.3 worked in the video contains a mistake, which is then carried over into Example 3.4. Email what the the mistake is and the correct results for an extra point.

5. Resources

6. Challenge 03

challenge03

Clearly write up your solutions to the questions in § 3.2, “The Nickel and the Battleship”. “Clearly” means that the reader doesn’t need to open the web page in order to figure out what the solution is about.

1. SI unit symbols are correctly written with an upright (not italic) font style. This is frequently ignored and can occasionaly cause unnecessary confusion in written work.
2. Magnetic field lines have no beginning or ending (!), this is a great mystery.
3. A nickel coin is an alloy of 75% Cu / 25% Ni. How many free electrons are contributed to the conduction band per atom for copper or nickel? This is not an easy question.
4. Thanks Archimedes!
5. Isn’t Chemistry just Physics applied to combinations of elements in space?